Tangent Theorem 1

In this post, we discuss a theorem on tangent of circles.  We show that if a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. In the figure below, line m is perpendicular to $latex OA$ at $latex A$. We show that it is a tangent to circle O.

Theorem

If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.

Given

Line $latex m$ is perpendicular to $latex OA$ at $latex A$.

Prove

Line $latex m$ is tangent to circle $latex O$ 

Proof

Let $latex B$ be another point on line $latex m$ other than $latex A$.

tangent theorem

Since $latex OA$ is perpendicular $latex m$, triangle $latex OAB$ is a right triangle with hypotenuse $latex OB$. This means that $latex OB > AB$ and $latex B$ must be on the exterior of the circle. Therefore $latex B$ cannot lie on the circle and $latex B$ is the only point of m that is on the circle. So, $latex m$ is a tangent to the circle.

The converse of this statement is also true (see proof).

1 thought on “Tangent Theorem 1

  1. Pingback: Tangent Theorem 2 - Proofs from The Book

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